Credit Risk Modeling

' Credit Risk Modeling $ 1 Credit Risk Modeling References: • An Introduction to Credit Risk Modeling by Bluhm, Overbeck and Wagner, Chapman & Hall, 2003 • Credit Risk by Duffie and Singleton, New Age International Publishers, 2005 • Credit Risk Modeling and Valuation: An Introduction, by Kay Giesecke, http://www.stanford.edu/dept/MSandE/people/faculty/giesecke/introduction.pdf, 2004 • Options, Futures, and Other Derivatives, Hull, Prentice Hall India & % ' Credit Risk Modeling $ 2 The Basics of Credit Risk Management • Loss Variable ˜ L = EAD × SEV × L = OU T ST + γ COM M • Exposure at Default (EAD) Basel Committee on banking supervision: 75% of off-balance sheet amount. Ex. Committed line of one billion, current outstandings 600 million, EAD = 600 + 75% × 400 = 900. • Loss Given Default (LGD) – Quality of collateral – Seniority of claim • L = 1D , P (D) = DP : Probability of Default – Calibration from market data, Ex. KMV Corp. – Calibration from ratings, Ex. Moodys, S & P, Fitch, CRISIL : Statistical tools + Soft factors – Ratings & → DP: Fit “curve” to RR vs average DP plot % = E[SEV ] ' Credit Risk Modeling $ 3 • Expected Loss (EL) • Unexpected Loss (UL) = EAD × ˜ Portfolio: LP F = • ELP F = • U L2 F = P m m i=1 m i=1 ˜ E[L] = EAD × LGD × DP = ˜ V (L) V ( SEV ) × DP 2 + LGD2 × DP (1 − DP ) EADi × SEVi × Li EADi × LGDi × DPi EADi × EADj × Cov(SEVi × Li , SEVj × Lj ) m i,j=1 • Constant Severities = i,j=1 EADi × EADj × LGDi × LGDj × DPi (1 − DPi )DPj (1 − DPj ) ρij & % ' Credit Risk Modeling $ qα ˜ qα : inf{q > 0 : P [LP F ≤ q] ≥ α} 4 • Value at Risk (VaR): • Economic Capital (ECα ) • Expected Shortfall: = qα − ELP F ˜ ˜ T CEα = E[LP F | LP F ≥qα ] T CEα − ELP F • Economic Capital based on Shortfall Risk: • Loss Distribution – Monte-Carlo Simulation – Analytical Approximation: Credit Risk+ • Today’s Industry Models – Credit Metrics and KMV-Model – Credit Risk+ – CreditPortfolio View – Dynamic Intensity Models & % ' Credit Risk Modeling $ 5 Credit Metrics and the KMV-Model • Asset Price Process: • Valuation Horizon: T Li = 1{A(i) j M (s) applies to all obligors in segment s. Some entries may turn out to be negative. Set equal to 0 and renormalize. & % ' Credit Risk Modeling $ 11 ms = αij (rs − 1) + mij ¯ ij • rs < 1 : expanding economy, lower possibility of downgrades and higher number of upgrades • rs = 1 : • rs > 1 : average macroeconomic scenario recession, downgrades more likely For each realization of the default probabilities, simulate the defaults and loss. Repeat simulation several times to generate the loss distribution. & % ' Credit Risk Modeling $ 12 CPV supports two modes of calibration: • CPV Macro: default and rating migrations are explained by a macroeconomic regression model. Macroeconomic model is calibrated by means of times series of empirical data. M Ys,t = ws,0 + j=1 ws,j Xs,j,t + t0 s,t , s,t 2 ∼ N (0, σs,t ) Xs,j,t = θj,0 + k=1 θj,k Xs,j,t−k + ηs,j,t 1 1 + exp(−ys,t ) ps,t = • CPV Direct: ps drawn from a gamma distribution. Need only to calibrate the two parameters of the gamma distribution for each s. ps can turn out to be larger than 1. & % ' Credit Risk Modeling $ 13 Dynamic Intensity Models • Basic Affine or Intensity Process dλ(t) = κ(θ − λ(t)) dt + σ λ(t) dB(t) + ∆J (t) • J (t) : pure jump process independent of the BM B(t) with jumps arriving according to a Poisson process with rate and jump sizes ∆J (t) ∼ exp(µ) • κ= mean-reversion rate; σ= m = θ + µ/κ ¯ long-run mean • Unconditional Default Probability diffusive volatility; q(t) = E e − Rt 0 λ(u) du • Correlated defaults λ i = Xc + Xi Xc , Xi basic affine processes with parameters (κ, θc , σ, µ, c ) and (κ, θi , σ, µ, i ) representing the common performance aspects and the idiosyncratic risk • λi : & basic affine process with parameters (κ, θc + θi , σ, µ, c + i) % ' Credit Risk Modeling $ 14 dXp (t) = d(Xc + Xi )(t) = κ(θp − Xp (t)) dt + σ Xp (t) dB p (t) + ∆J p (t), p = c, i κ((θc + θi ) − (Xc + Xi )(t)) dt +σ( Xc (t) dB c (t) + c Xt dB c (t) + c i Xt + Xt Xi (t) dB i (t)) + ∆(J c + J i )(t) i Xt dB i (t) c i Xt + Xt dWt = d(Xc + Xi )(t) = κ((θc + θi ) − (Xc + Xi )(t)) dt + σ +∆(J c + J i )(t) (X c + X i )(t) dW (t) Conditioned on a realization of λi (t), 0 ≤ t ≤ T , the default time of obligor i is the first arrival in a non-homogenous Poisson process with rate λi (·) Conditional Probability of No Default & = exp(− T 0 λ(s)ds) % ' Credit Risk Modeling $ 15 The Credit Risk+ Model • Introduced in 1997 by CSFB • Actuarial Model • One of the most widely used credit portfolio models • Advantages: – Loss Distribution can be computed analytically – Requires no Monte-Carlo Simulations – Explicit Formulas for Obligor Risk Contributions • Numerically stable computational procedure (Giese, 2003) & % ' Credit Risk Modeling $ 16 The Standard CR+ Model • Choose a suitable basic unit of currency • Adjusted exposure of obligor A, • Smaller number of Exposure Bands • pA expected default probability L= A ∆L νA = EA /∆L • The total portfolio loss • NA ∈ Z+ νA NA . ∞ n=0 Default of obligor A G(z) = P (L = n) z n . • PGF of the Loss Distribution & % ' Credit Risk Modeling $ 17 • Apportion Obligor Risk among K Sectors (Industry, Country) by choosing K A A numbers gk such that k=1 gk = 1. • Sectoral Default Rates represented by non-negative variables γk E(γk ) = 1, Cov(γk , γl ) = σkl k=l k = 1, ...., K. • Standard CR+ Model assumes σkl = 0, • Relating Obligor default rates to sectoral default rates K pA (γ) = pA k=1 A gk γk , • pA (γ) default rate conditional on the sector default rates γ = (γ1 , . . . , γK ). • Specific Sector: γ0 ≡ 1. Captures Idosyncratic Risk & % ' Credit Risk Modeling $ 18 • Conditional on γ default variables NA assumed to be independent Poisson • Main Criticism of CR+ Model. Not Fair – pA = 0.1 • Conditional PGF Gγ (z) = exp( k=1 → P (NA = 2) = 0.0045 – Need not assume NA is Poisson, but Bernoulli K γk Pk (z)), Pk (z) = A M A gk pA (z νA − 1)   {νA =m}  A gk pA  (z m − 1) = m=1 • Number of defaults in any exposure band is Poisson & % ' Credit Risk Modeling $ 19 • Default correlation between obligors arise only through their dependence on the common set of sector default rates • Unconditional PGF of Loss Distribution K G(z) = E (exp( γ k=1 γk Pk (z))) = Mγ (T = P (z)) • MGF of Univariate Gamma Distribution with Mean 1 and Variance σkk is − σ1 (1 − σkk tk ) kk K GCR+ (z) = exp − k=1 1 log(1 − σkk Pk (z)) σkk • Giese(2003): Numerically Stable Fast Recursion Scheme & % ' Credit Risk Modeling $ 20 The Compound Gamma CR+ Model (Giese, 2003) • Introduce sectoral correlations via common scaling factor S • Conditional on S γK is Gamma distributed with shape parameter αk > 0, and constant scale parameter βk . αk (S) = Sαk , ˆ • S follows Gamma with E[S] = 1 and V ar(S) = σ 2 . ˆ • 1 = Eγk = αk βk • σkl = δkl βk + σ 2 ˆ • Uniform Level of Cross Covariance Structure. CG Mγ (T ) ⇒ K k=1 Distortion of Correlation 1 log(1 − βk tk ) βk 1 = exp − 2 log 1 + σ 2 ˆ σ ˆ • Calibration Problems & % ' Credit Risk Modeling $ 21 The Two Stage CR+ Model (SKI, AD) • Y1 , . . . , YN : Common set of Uncorrelated Risk Drivers N γk = i=1 aki Yi • Yi ∼ Gamma with mean 1 and variance Vii • Principle Component Analysis of Macroeconomic Variables • Factor Analysis & % ' Credit Risk Modeling $ 22 K K N G(z) = E γ (exp( k=1 N γk Pk (z))) = E Y (exp( K ( k=1 i=1 aki Yi ) Pk (z))) = E Y (exp( N ( i=1 k=1 aki Pk (z)) Yi )) = E (exp( Y i=1 Yi Qi (z))) = MY (T = Q(z)) K Qi (z) = k=1 N aki Pk (z) 1 log(1 − σii Qi (z)) σii G(z) = exp − i=1 & % ' Credit Risk Modeling $ 23 Model Comparison • Giese (2003) had pointed out deficiencies in the earlier attempt to incorporate correlations due to Burgisser et al • We compare the compound gamma and the two stage gamma models • Test portfolio made up of K = 12 sectors, each containing 3,000 obligors • Obligors in sectors 1 to 10 belong in equal parts to one of three classes with adjusted exposures E1 = 1, E2 = 2.5, and E3 = 5 monetary units and respective default probabilities p1 = 5.5%, p2 = .8%, p3 = .2%. • For the three obligor classes in sectors 11 and 12, we assume the same default rates but twice as large exposures (E1 = 2, E2 = 5, E3 = 10) • σkk = 0.04, k = 1, . . . , 10 σ11,11 = σ12,12 = 0.49 • Correlation between sectors 11 and 12 is 0.5 whereas correlations between all the other sectors are set equal to 0 • γi = Yi , i = 1, . . . , 11, γ12 = 0.5(Y11 + Y12 ), with V ar(Y11 ) = 0.49 V ar(Y12 ) = 1.47, and Var(Yi ) = 0.04 for i = 1, . . . , 10 & % ' Credit Risk Modeling $ 24 Standard CR+ Expected Loss Std Deviation 99% Quantile 99.5% Quantile 99.9% Quantile 1% 0.15% 1.42% 1.49% 1.64% Compound Gamma Model 1% 0.17% 1.48% 1.55% 1.71% Two-Stage Model 1% 0.17% 1.53% 1.62% 1.84% Table 1: Comparison of the loss distributions from the standard CR+ , compound gamma and two stage models for the test portfolio in example 1. All loss statistics are quoted as percentage of the total adjusted exposure. • σ 2 = 0.013. This translates to a much lower correlation of 0.0265 (instead of ˆ 0.5) between sectors 11 and 12 & % ' Credit Risk Modeling $ 25 Risk Contributions • Value at Risk VAR • Economic Capital • Expected Shortfall q q − E[L] q] E[L|L ≥ • Quantile Contribution QCA K k ( q −νA ) Gk (z) k=1 gA D D( q ) G(z) QCA = νA E(NA |L = Gk (z) = ∂ ∂tk MY q) = pA νA (T = Q(z)) N Gk (z) = G(z) i=1 N aki 1 − σii Qi (z) ak,i exp(−log(1 − σii Qi (z)))). = G(z) ( i=1 & % ' Credit Risk Modeling $ 26 Sector 1, 2 3, . . . , 10 11, 12 CR+ 24.25% 0.37% 24.25 % Compound Gamma Model 21.71% 1.64 % 21.71% Two-Stage Model 27.42% 0.2 % 21.59 % Table 2: Aggregated risk contributions (in percent). Contributions to the loss variance for the risk-adjusted breakdown of VaR (on a 99.9% confidence level). • Compound gamma model can’t pick up differing correlations among sectors that are otherwise similar. & %